In this work, a novel approach for the solution of the inverse conductivity problem from one and multiple boundary measurements has been developed on the basis of the implication of the framework of BV functions. The space of the functions of bounded variation is recommended here as the most appropriate functional space hosting the conductivity profile under reconstruction. For the numerical investigation of the inversion of the inclusion problem, we propose and implement a suitable minimization scheme of an enriched-constructed herein-functional, by exploiting the inner structure of BV space. Finally, we validate and illustrate our theoretical results with numerical experiments.KEYWORDS boundary value problems for second-order elliptic equations, inverse problems MSC CLASSIFICATION 35J25; 35R30Math Meth Appl Sci. 2020;43:5032-5072. wileyonlinelibrary.com/journal/mma

Recently in Charalambopoulos et al. (2020), we presented a methodology aiming at reconstructing bounded total variation ($TV$) conductivities via a technique simulating the so-called half-quadratic minimization approach, encountered in Aubert & Kornprobst (2002, Mathematical Problems in Image Processing. New York, NY: Springer). The method belongs to a duality framework, in which the auxiliary function $\omega (x)$ was introduced, offering a tool for smoothing the members of the admissible set of conductivity profiles. The dual variable $\omega (x)$, in that approach, after every external update, served in the formation of an intermediate optimization scheme, concerning exclusively the sought conductivity $\alpha (x)$. In this work, we develop a novel investigation stemming from the previous approach, having though two different fundamental components. First, we do not detour herein the $BV$-assumption on the conductivity profile, which means that the functional under optimization contains the $TV$ of $\alpha (x)$ itself. Secondly, the auxiliary dual variable $\omega (x)$ and the conductivity $\alpha (x)$ acquire an equivalent role and concurrently, a parallel pacing in the minimization process. A common characteristic between these two approaches is that the function $\omega (x)$ is an indicator of the conductivity’s ‘jump’ set. A fortiori, this crucial property has been ameliorated herein, since the reciprocal role of the elements of the pair $(\alpha ,\omega )$ offers a self-monitoring structure very efficient to the minimization descent.

Recently in [V. Markaki, D. Kourounis and A. Charalambopoulos, A dual self-monitored reconstruction scheme on the TV \mathrm{TV} -regularized inverse conductivity problem, IMA J. Appl. Math. 86 2021, 3, 604–630], a novel reconstruction scheme has been developed for the solution of the inclusion problem in the inverse conductivity problem on the basis of a weighted self-guided regularization process generalizing the total variation approach. The present work extends this concept by implementing and investigating its applicability in the two-dimensional elasticity setting. To this end, we employ the framework of the reconstruction of linear and isotropic elastic structures described by their Lamé parameters. Numerical examples of increasingly challenging geometric complexities illustrate the enhanced accuracy and efficiency of the method.